Analysis on Manifolds with Singularities
Description
This exposition introduces new developments of pseudo-differential operators on manifolds with singularities such as conical points, edges, or higher corners, and recent results in these fields. It presents some crucial parts of recent inventions in singular analysis which contribute to pseudo-differential structures and establishes new insight to building up singular operators in corresponding operator algebras with symbolic structures.
Researchers in the field will find the presented techniques in this book a valuable resource and inspiration for their own projects.
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Der-Chen Chang is a professor and McDevitt Chair in Mathematics and Computer Science in the Department of Mathematics and Statistics. He also serves as senior advisor to the Provost for China initiatives.
Bert-Wolfgang Schulze is Emeritus Professor at the University of Potsdam, Germany. He has authored 15 books and is an editor for various journals and book series in the fields of Partial Differential Equations and Pseudo-Differential Operators. He is the initiator of the yearly conference on "Microlocal and global analysis, interactions with geometry" in Potsdam.
Content
Introduction.- 1. Pseudo-differential operators - basics and new techniques.- 1.1. Elements of the local calculus.- 1.2. Operators on manifolds.- 1.3. Twisted symbolic estimates.- 1.4. Edge Sobolev spaces.- 1.5. Explicit solutions to Dirichlet and Neumann problems.- 1.6. Hölder and Lp estimates for coercive boundary value problems.- 2. Operators of Fuchs Type.- 2.1. Degenerate Operators and Singular Manifolds.- 2.2. Kernel Cut-Off.- 2.3. Symbols with Distributional Asymptotic Densities.- 2.4. Conical singularities in stretched descripion.- 2.5. The cone calculus.- 3. Manifolds with edges.- 3.1. Manifolds with edge and weighted Sobolev spaces.- 3.2. Edge spaces with asymptotics.- 3.3. Edge operators.- 4. Edge Operators with Trace and Potential Conditions.- 4.1. Basic Tools.- 4.2. Group Actions and Weighted Cone and Edge Spaces.- 4.3. Operators in Edge Spaces.- 4.4. Notes on Pseudo-Differential Boundary Problems.- 4.5. The Laplace-Beltrami Operator on a Wedge.- 4.6. The Global Ellipticity of Edge Problems.- 4.7. Local Calculus Close to the Edge.- 4.8. Global Ellipticity with Edge Conditions.- 5. The Geometry of Singularities.- 5.1. The Case of Conical Singularities or Edges.- 5.2. Higher Singularities.- 5.3. Iterated Edge Spaces.- 5.4. Distribution spaces and cone operators.- Bibliography.